Properties

Label 60.3.c.a
Level $60$
Weight $3$
Character orbit 60.c
Analytic conductor $1.635$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [60,3,Mod(31,60)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(60, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("60.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 60.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.63488158616\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.85100625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 2x^{6} + x^{5} + 3x^{4} + 2x^{3} - 8x^{2} - 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + \beta_{4} q^{3} + (\beta_{5} + \beta_{3} + \beta_{2} + 1) q^{4} - \beta_{2} q^{5} + (\beta_{4} + \beta_{3} - \beta_1 - 1) q^{6} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_1) q^{7} + (2 \beta_{7} + 2 \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 - 4) q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + \beta_{4} q^{3} + (\beta_{5} + \beta_{3} + \beta_{2} + 1) q^{4} - \beta_{2} q^{5} + (\beta_{4} + \beta_{3} - \beta_1 - 1) q^{6} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_1) q^{7} + (2 \beta_{7} + 2 \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 - 4) q^{8} - 3 q^{9} + ( - \beta_{7} + \beta_{3} - \beta_{2} + 2) q^{10} + (2 \beta_{6} - 2 \beta_{5} - 2 \beta_{3}) q^{11} + (\beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2}) q^{12} + (\beta_{7} - 3 \beta_{6} - 3 \beta_{5} - \beta_{4} - 5 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 4) q^{13} + ( - 2 \beta_{7} + 2 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 4) q^{14} + ( - \beta_{7} - \beta_1) q^{15} + (3 \beta_{7} - 3 \beta_{6} - 4 \beta_{5} - \beta_{4} + \beta_{2} + 3 \beta_1 + 7) q^{16} + ( - \beta_{7} - \beta_{6} - 5 \beta_{5} + \beta_{4} + \beta_{3} - 6 \beta_{2} + \beta_1 + 4) q^{17} - 3 \beta_{5} q^{18} + ( - 5 \beta_{7} - \beta_{6} + 3 \beta_{5} - 7 \beta_{4} + \beta_{3} - 3 \beta_1) q^{19} + ( - \beta_{7} + \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - 6) q^{20} + (\beta_{7} + \beta_{6} + 5 \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 - 10) q^{21} + ( - 2 \beta_{7} - 2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 8) q^{22} + (7 \beta_{7} - \beta_{6} - 9 \beta_{5} + 9 \beta_{4} + \beta_{3} - 3 \beta_1) q^{23} + (\beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - 5 \beta_{2} - \beta_1 + 3) q^{24} + 5 q^{25} + ( - 2 \beta_{7} - 4 \beta_{6} + 8 \beta_{4} - 6 \beta_{3} + 6 \beta_{2} - 4) q^{26} - 3 \beta_{4} q^{27} + (2 \beta_{7} + 6 \beta_{6} - 4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 2) q^{28} + ( - 4 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} + 4 \beta_{4} + 12 \beta_{3} + \cdots + 12) q^{29}+ \cdots + ( - 6 \beta_{6} + 6 \beta_{5} + 6 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 10 q^{4} - 6 q^{6} - 20 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} + 10 q^{4} - 6 q^{6} - 20 q^{8} - 24 q^{9} + 10 q^{10} + 16 q^{13} - 20 q^{14} + 34 q^{16} - 12 q^{18} - 40 q^{20} - 48 q^{21} + 68 q^{22} + 18 q^{24} + 40 q^{25} - 36 q^{26} + 28 q^{28} + 64 q^{29} - 76 q^{32} - 92 q^{34} - 30 q^{36} - 112 q^{37} - 40 q^{38} - 10 q^{40} - 16 q^{41} + 108 q^{42} + 172 q^{44} + 152 q^{46} + 48 q^{48} - 56 q^{49} + 20 q^{50} - 128 q^{52} + 352 q^{53} + 18 q^{54} + 116 q^{56} + 144 q^{57} - 204 q^{58} + 30 q^{60} - 176 q^{61} - 56 q^{62} - 110 q^{64} - 80 q^{65} + 108 q^{66} - 184 q^{68} - 96 q^{69} - 60 q^{70} + 60 q^{72} - 240 q^{73} + 132 q^{74} - 24 q^{76} - 288 q^{77} - 240 q^{78} - 80 q^{80} + 72 q^{81} + 40 q^{82} - 36 q^{84} + 160 q^{85} - 200 q^{86} + 140 q^{88} + 80 q^{89} - 30 q^{90} + 144 q^{92} + 144 q^{93} - 96 q^{94} - 174 q^{96} + 432 q^{97} + 660 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 2x^{6} + x^{5} + 3x^{4} + 2x^{3} - 8x^{2} - 8x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} - \nu^{6} + 7\nu^{4} - 5\nu^{3} - 4\nu^{2} + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{7} - \nu^{6} - 4\nu^{5} - 5\nu^{4} - 5\nu^{3} + 16\nu^{2} - 8\nu - 24 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - \nu^{6} - 2\nu^{5} + \nu^{4} + 3\nu^{3} + 2\nu^{2} - 8\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} - \nu^{6} + 4\nu^{5} + 3\nu^{4} - 5\nu^{3} - 8\nu^{2} + 8\nu + 40 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} - \nu^{6} + 4\nu^{5} + 3\nu^{4} - 5\nu^{3} - 24\nu^{2} + 24\nu + 56 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + \nu^{6} - 2\nu^{5} - \nu^{4} + \nu^{3} + 4\nu^{2} - 4\nu - 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{7} + 3\nu^{6} - 12\nu^{5} - 9\nu^{4} + 15\nu^{3} + 32\nu^{2} - 8\nu - 80 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} - 3\beta_{5} + 3\beta_{4} - \beta_{3} + \beta _1 + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 4\beta_{2} - \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} - \beta_{6} - 3\beta_{5} - \beta_{4} - \beta_{3} - 4\beta_{2} + 5\beta _1 + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} - 3\beta_{6} - 9\beta_{5} - 3\beta_{4} - 3\beta_{3} - 4\beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{7} + 5\beta_{6} - 3\beta_{5} - \beta_{4} - 5\beta_{3} - 8\beta_{2} + \beta _1 + 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -\beta_{7} - 3\beta_{6} - \beta_{5} - 23\beta_{4} - 3\beta_{3} + 3\beta _1 + 24 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−1.34966 + 0.422403i
−1.34966 0.422403i
1.40906 + 0.120653i
1.40906 0.120653i
−0.600040 1.28061i
−0.600040 + 1.28061i
1.04064 0.957636i
1.04064 + 0.957636i
−1.99281 0.169449i 1.73205i 3.94257 + 0.675358i −2.23607 −0.293494 + 3.45165i 12.3959i −7.74236 2.01392i −3.00000 4.45606 + 0.378899i
31.2 −1.99281 + 0.169449i 1.73205i 3.94257 0.675358i −2.23607 −0.293494 3.45165i 12.3959i −7.74236 + 2.01392i −3.00000 4.45606 0.378899i
31.3 0.438172 1.95141i 1.73205i −3.61601 1.71011i 2.23607 −3.37994 0.758935i 6.33166i −4.92155 + 6.30701i −3.00000 0.979781 4.36349i
31.4 0.438172 + 1.95141i 1.73205i −3.61601 + 1.71011i 2.23607 −3.37994 + 0.758935i 6.33166i −4.92155 6.30701i −3.00000 0.979781 + 4.36349i
31.5 1.67986 1.08539i 1.73205i 1.64388 3.64660i 2.23607 1.87994 + 2.90961i 0.596540i −1.19648 7.91002i −3.00000 3.75629 2.42700i
31.6 1.67986 + 1.08539i 1.73205i 1.64388 + 3.64660i 2.23607 1.87994 2.90961i 0.596540i −1.19648 + 7.91002i −3.00000 3.75629 + 2.42700i
31.7 1.87477 0.696577i 1.73205i 3.02956 2.61185i −2.23607 −1.20651 3.24721i 5.46770i 3.86039 7.00695i −3.00000 −4.19212 + 1.55759i
31.8 1.87477 + 0.696577i 1.73205i 3.02956 + 2.61185i −2.23607 −1.20651 + 3.24721i 5.46770i 3.86039 + 7.00695i −3.00000 −4.19212 1.55759i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.3.c.a 8
3.b odd 2 1 180.3.c.b 8
4.b odd 2 1 inner 60.3.c.a 8
5.b even 2 1 300.3.c.d 8
5.c odd 4 2 300.3.f.b 16
8.b even 2 1 960.3.e.c 8
8.d odd 2 1 960.3.e.c 8
12.b even 2 1 180.3.c.b 8
15.d odd 2 1 900.3.c.u 8
15.e even 4 2 900.3.f.f 16
20.d odd 2 1 300.3.c.d 8
20.e even 4 2 300.3.f.b 16
24.f even 2 1 2880.3.e.j 8
24.h odd 2 1 2880.3.e.j 8
60.h even 2 1 900.3.c.u 8
60.l odd 4 2 900.3.f.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.c.a 8 1.a even 1 1 trivial
60.3.c.a 8 4.b odd 2 1 inner
180.3.c.b 8 3.b odd 2 1
180.3.c.b 8 12.b even 2 1
300.3.c.d 8 5.b even 2 1
300.3.c.d 8 20.d odd 2 1
300.3.f.b 16 5.c odd 4 2
300.3.f.b 16 20.e even 4 2
900.3.c.u 8 15.d odd 2 1
900.3.c.u 8 60.h even 2 1
900.3.f.f 16 15.e even 4 2
900.3.f.f 16 60.l odd 4 2
960.3.e.c 8 8.b even 2 1
960.3.e.c 8 8.d odd 2 1
2880.3.e.j 8 24.f even 2 1
2880.3.e.j 8 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(60, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 4 T^{7} + 3 T^{6} + 16 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + 224 T^{6} + 12032 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$11$ \( (T^{4} + 208 T^{2} + 10496)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 8 T^{3} - 472 T^{2} + 5792 T - 12464)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 424 T^{2} + 3840 T - 8816)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 1696 T^{6} + \cdots + 6544162816 \) Copy content Toggle raw display
$23$ \( T^{8} + 3616 T^{6} + \cdots + 101419319296 \) Copy content Toggle raw display
$29$ \( (T^{4} - 32 T^{3} - 2152 T^{2} + \cdots + 1334416)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 5408 T^{6} + \cdots + 59895709696 \) Copy content Toggle raw display
$37$ \( (T^{4} + 56 T^{3} - 1528 T^{2} + \cdots - 244784)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 8 T^{3} - 1800 T^{2} + \cdots + 87184)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 10816 T^{6} + \cdots + 33624411406336 \) Copy content Toggle raw display
$47$ \( T^{8} + 8032 T^{6} + \cdots + 1056981385216 \) Copy content Toggle raw display
$53$ \( (T^{4} - 176 T^{3} + 9752 T^{2} + \cdots - 478064)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 4896 T^{6} + \cdots + 173909016576 \) Copy content Toggle raw display
$61$ \( (T^{4} + 88 T^{3} - 2536 T^{2} + \cdots - 2142704)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 16064 T^{6} + \cdots + 281086590976 \) Copy content Toggle raw display
$71$ \( T^{8} + 13952 T^{6} + \cdots + 16079971680256 \) Copy content Toggle raw display
$73$ \( (T^{4} + 120 T^{3} - 1576 T^{2} + \cdots + 4962064)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 41888 T^{6} + \cdots + 31\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{8} + 36928 T^{6} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} - 40 T^{3} - 20584 T^{2} + \cdots + 70652944)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 216 T^{3} + 5880 T^{2} + \cdots - 59281776)^{2} \) Copy content Toggle raw display
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